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Properties

Label 2-770-55.43-c1-0-23
Degree $2$
Conductor $770$
Sign $0.845 - 0.533i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (0.436 + 0.436i)3-s + 1.00i·4-s + (0.925 − 2.03i)5-s + 0.617i·6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 2.61i·9-s + (2.09 − 0.784i)10-s + (−0.0914 + 3.31i)11-s + (−0.436 + 0.436i)12-s + (2.24 − 2.24i)13-s + 1.00i·14-s + (1.29 − 0.484i)15-s − 1.00·16-s + (0.882 + 0.882i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.251 + 0.251i)3-s + 0.500i·4-s + (0.413 − 0.910i)5-s + 0.251i·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.873i·9-s + (0.662 − 0.248i)10-s + (−0.0275 + 0.999i)11-s + (−0.125 + 0.125i)12-s + (0.622 − 0.622i)13-s + 0.267i·14-s + (0.333 − 0.125i)15-s − 0.250·16-s + (0.214 + 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37545 + 0.686758i\)
\(L(\frac12)\) \(\approx\) \(2.37545 + 0.686758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.925 + 2.03i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.0914 - 3.31i)T \)
good3 \( 1 + (-0.436 - 0.436i)T + 3iT^{2} \)
13 \( 1 + (-2.24 + 2.24i)T - 13iT^{2} \)
17 \( 1 + (-0.882 - 0.882i)T + 17iT^{2} \)
19 \( 1 - 5.11T + 19T^{2} \)
23 \( 1 + (-4.55 - 4.55i)T + 23iT^{2} \)
29 \( 1 - 3.89T + 29T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 + (-1.13 + 1.13i)T - 37iT^{2} \)
41 \( 1 + 2.41iT - 41T^{2} \)
43 \( 1 + (6.96 - 6.96i)T - 43iT^{2} \)
47 \( 1 + (-6.01 + 6.01i)T - 47iT^{2} \)
53 \( 1 + (8.55 + 8.55i)T + 53iT^{2} \)
59 \( 1 - 8.83iT - 59T^{2} \)
61 \( 1 - 6.74iT - 61T^{2} \)
67 \( 1 + (-1.48 + 1.48i)T - 67iT^{2} \)
71 \( 1 + 3.65T + 71T^{2} \)
73 \( 1 + (2.44 - 2.44i)T - 73iT^{2} \)
79 \( 1 + 8.00T + 79T^{2} \)
83 \( 1 + (3.86 - 3.86i)T - 83iT^{2} \)
89 \( 1 + 5.55iT - 89T^{2} \)
97 \( 1 + (2.55 - 2.55i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.13591623545164507495499815497, −9.381491595046226355218997918403, −8.744708087967528760567658620264, −7.81930591803765568737419711625, −6.87766028444707681858860767967, −5.72179232342129279054177796228, −5.14758964324949483073066184792, −4.09479140645365165192881710297, −3.05796552386345137359698236197, −1.38078399697436254006967133189, 1.43175909632736712865891014585, 2.69483106194232460513894799051, 3.46519253727148053492876586536, 4.80839598164001698311184895609, 5.74926949659673286907046300879, 6.70244113877885384946964999546, 7.54586092553164483277761865728, 8.582736856152295151510067386660, 9.526930512865200389651440947122, 10.59730320006642210046302817505

Graph of the $Z$-function along the critical line